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All the ideas for 'Classical Cosmology (frags)', 'The Later Works (17 vols, ed Boydston)' and 'A Tour through Mathematical Logic'

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31 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Philosophy is the study and criticsm of cultural beliefs, to achieve new possibilities [Dewey]
     Full Idea: Philosophy is criticism of the influential beliefs that underlie culture, tracking them to their generating conditions and results, and considering their mutual compatibility. This terminates in a new perspective, which leads to new possibilities.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 6:19), quoted by David Hildebrand - Dewey Intro
     A reaction: [compressed] This would make quite a good manifesto for French thinkers of the 1960s. Foucault could hardly disagree. An excellent idea.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'tautology' must include connectives [Wolf,RS]
     Full Idea: 'For every number x, x = x' is not a tautology, because it includes no connectives.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.2)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
     Full Idea: Deduction Theorem: If T ∪ {P} |- Q, then T |- (P → Q). This is the formal justification of the method of conditional proof (CPP). Its converse holds, and is essentially modus ponens.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
     Full Idea: Universal Generalization: If we can prove P(x), only assuming what sort of object x is, we may conclude ∀xP(x) for the same x.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
     A reaction: This principle needs watching closely. If you pick one person in London, with no presuppositions, and it happens to be a woman, can you conclude that all the people in London are women? Fine in logic and mathematics, suspect in life.
Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
     Full Idea: Universal Specification: from ∀xP(x) we may conclude P(t), where t is an appropriate term. If something is true for all members of a domain, then it is true for some particular one that we specify.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
     Full Idea: Existential Generalization (or 'proof by example'): From P(t), where t is an appropriate term, we may conclude ∃xP(x).
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.3)
     A reaction: It is amazing how often this vacuous-sounding principles finds itself being employed in discussions of ontology, but I don't quite understand why.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
     Full Idea: Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3)
     A reaction: A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
     Full Idea: The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2)
     A reaction: This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Liberalism should improve the system, and not just ameliorate it [Dewey]
     Full Idea: Liberalism must become radical in the sense that, instead of using social power to ameliorate the evil consequences of the existing system, it shall use social power to change the system.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:287), quoted by David Hildebrand - Dewey 4 'Dewey'
     A reaction: Conservative liberals ask what people want, and try to give it to them. Radical liberals ask what people actually need, and try to make it possible. The latter is bound to be a bit paternalistic, but will probably create a better world.
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
     Full Idea: One of the most appealing features of first-order logic is that the two 'turnstiles' (the syntactic single |-, and the semantic double |=), which are the two reasonable notions of logical consequence, actually coincide.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: In the excitement about the possibility of second-order logic, plural quantification etc., it seems easy to forget the virtues of the basic system that is the target of the rebellion. The issue is how much can be 'expressed' in first-order logic.
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
     Full Idea: The 'completeness' of first order-logic does not mean that every sentence or its negation is provable in first-order logic. We have instead the weaker result that every valid sentence is provable.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: Peter Smith calls the stronger version 'negation completeness'.
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
Model theory reveals the structures of mathematics [Wolf,RS]
     Full Idea: Model theory helps one to understand what it takes to specify a mathematical structure uniquely.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.1)
     A reaction: Thus it is the development of model theory which has led to the 'structuralist' view of mathematics.
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
     Full Idea: A 'structure' in model theory has a non-empty set, the 'universe', as domain of variables, a subset for each 'relation', some 'functions', and 'constants'.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.2)
Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
     Full Idea: Model theory uses set theory to show that the theorem-proving power of the usual methods of deduction in mathematics corresponds perfectly to what must be true in actual mathematical structures.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
     A reaction: That more or less says that model theory demonstrates the 'soundness' of mathematics (though normal arithmetic is famously not 'complete'). Of course, he says they 'correspond' to the truths, rather than entailing them.
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
     Full Idea: The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)
     A reaction: On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223).
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
     Full Idea: An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4)
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
     Full Idea: The Löwenheim-Skolem-Tarski theorem demonstrates a serious limitation of first-order logic, and is one of primary reasons for considering stronger logics.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.7)
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
     Full Idea: It is valuable to know that a theory is complete, because then we know it cannot be strengthened without passing to a more powerful language.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.5)
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
     Full Idea: Deductive logic, including first-order logic and other types of logic used in mathematics, is 'monotonic'. This means that we never retract a theorem on the basis of new givens. If T|-φ and T⊆SW, then S|-φ. Ordinary reasoning is nonmonotonic.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 1.7)
     A reaction: The classic example of nonmonotonic reasoning is the induction that 'all birds can fly', which is retracted when the bird turns out to be a penguin. He says nonmonotonic logic is a rich field in computer science.
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
     Full Idea: Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
     A reaction: He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Modern mathematics has unified all of its objects within set theory [Wolf,RS]
     Full Idea: One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
     A reaction: His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses.
11. Knowledge Aims / A. Knowledge / 1. Knowledge
Knowledge is either the product of competent enquiry, or it is meaningless [Dewey]
     Full Idea: Knowledge, as an abstract term, is a name for the product of competent enquiries. Apart from this relation, its meaning is so empty that any content or filling may be arbitrarily poured into it.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 12:16), quoted by David Hildebrand - Dewey 2 'Knowledge'
     A reaction: What is the criterion of 'competent'? Danger of tautology, if competent enquiry is what produces knowledge.
11. Knowledge Aims / B. Certain Knowledge / 1. Certainty
The quest for certainty aims for peace, and avoidance of the stress of action [Dewey]
     Full Idea: The quest for certainty is a quest for a peace which is assured, an object which is unqualified by risk and the shadow of fear which action costs.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 4:7), quoted by David Hildebrand - Dewey 2 'Intro'
     A reaction: This is a characteristic pragmatist account. I think Dewey and Peirce offer us the correct attitude to certainty. It is just not available to us, and can only be a delusion. That doesn't mean we don't know anything, however!
11. Knowledge Aims / B. Certain Knowledge / 3. Fallibilism
No belief can be so settled that it is not subject to further inquiry [Dewey]
     Full Idea: The attainment of settled beliefs is a progressive matter; there is no belief so settled as not to be exposed to further inquiry.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 12:16), quoted by David Hildebrand - Dewey 2 'Knowledge'
     A reaction: A nice pragmatist mantra, but no scientists gets a research grant to prove facts which have been securely established for a very long time. It is neurotic to keep returning to check that you have locked your front door. Dewey introduced 'warranted'.
15. Nature of Minds / A. Nature of Mind / 1. Mind / a. Mind
Mind is never isolated, but only exists in its interactions [Dewey]
     Full Idea: Mind is primarily a verb. ...Mind never denotes anything self-contained, isolated from the world of persons and things, but is always used with respect to situations, events, objects, persons and groups.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 10:267), quoted by David Hildebrand - Dewey 1 'emerge'
     A reaction: I strongly agree with the idea that mind is a process, not a thing. Certain types of solitary introspection don't seem to quite fit his account, but in general he is right.
24. Political Theory / D. Ideologies / 6. Liberalism / a. Liberalism basics
Liberals aim to allow individuals to realise their capacities [Dewey]
     Full Idea: Liberalism is committed to …the liberation of individuals so that realisation of their capacities may be the law of their life.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 11:41), quoted by David Hildebrand - Dewey 4 'Dewey'
     A reaction: Capacity expression as the main aim of politics is precisely the idea developed more fully in modern times by Amartya Sen and Martha Nussbaum. It strikes me as an excellent proposal. Does it need liberalism, or socialism?
24. Political Theory / D. Ideologies / 7. Communitarianism / a. Communitarianism
The things in civilisation we prize are the products of other members of our community [Dewey]
     Full Idea: The things in civilisation we most prize are not of ourselves. They exist by grace of the doings and sufferings of the continuous human community in which we are a link
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:57), quoted by David Hildebrand - Dewey 7 'Reconstruct'
     A reaction: Dewey defends liberalism, but he has strong communitarian tendencies. What is the significance of an enduring community losing touch with its own achievements?
27. Natural Reality / E. Cosmology / 1. Cosmology
Is the cosmos open or closed, mechanical or teleological, alive or inanimate, and created or eternal? [Robinson,TM, by PG]
     Full Idea: The four major disputes in classical cosmology were whether the cosmos is 'open' or 'closed', whether it is explained mechanistically or teleologically, whether it is alive or mere matter, and whether or not it has a beginning.
     From: report of T.M. Robinson (Classical Cosmology (frags) [1997]) by PG - Db (ideas)
     A reaction: A nice summary. The standard modern view is closed, mechanistic, inanimate and non-eternal. But philosophers can ask deeper questions than physicists, and I say we are entitled to speculate when the evidence runs out.
28. God / A. Divine Nature / 2. Divine Nature
'God' is an imaginative unity of ideal values [Dewey]
     Full Idea: 'God' represents a unification of ideal values that is essentially imaginative in origin.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:29), quoted by David Hildebrand - Dewey 7 'Construct'
     A reaction: This seems to have happened when a flawed God like Zeus is elevated to be the only God, and is given supreme power and wisdom.
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
We should try attaching the intensity of religious devotion to intelligent social action [Dewey]
     Full Idea: One of the few experiments in the attachment of emotion to ends that mankind has not tried is that of devotion (so intense as to be religious) to intelligence as a force in social action.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:53), quoted by David Hildebrand - Dewey 7 'Intro'
     A reaction: An interesting thought that religious emotions such as devotion are so distinctive that they can be treated as valuable, even in the absence of belief. He seems to be advocating Technocracy.
Religions are so shockingly diverse that they have no common element [Dewey]
     Full Idea: There is only a multitude of religions …and the differences between them are so great and so shocking that any common element that can be extracted is meaningless.
     From: John Dewey (The Later Works (17 vols, ed Boydston) [1930], 9:7), quoted by David Hildebrand - Dewey 7 'Construct'
     A reaction: Religion is for Dewey what a game was for Wittgenstein, as an anti-essentialist example. I would have thought that they all involved some commitment to a realm of transcendent existence.